Research #5: Finite-Alphabet Robustness Under Phase Uncertainty — Spectrum Forcing and Joint Observation–Energy–Information Frontiers

Research #5 unifies the preceding constellation-geometry, service-feasibility, deployment-accounting, and exact multiuser hierarchy studies into a theory of architecture-dependent resource costs under independent carrier-phase uncertainty. This record releases manuscript version 1.7 (61 pages), dated 30 September 2026. An exact finite two-user test determines all-user geometric-hierarchy optimality within specified spectrum classes. Analytic proofs and exact arithmetic certificates cover 33 finite square-QAM orders; the all-order geometric QAM conjecture and unrestricted finite-spectrum classification remain open. Norm forcing transfers to positive quadratic power costs, and heterogeneous uniform PAM has a complete phase-adversarial mixed-radix optimum under actual mean-power normalization. Established constellation constructions and classical information-theoretic ingredients are acknowledged explicitly. For K uniform lambda-PAM users partitioned into L orthogonal scalar groups, the paper proves an exact geometric energy frontier and matching-order received signal SNR Theta(L * lambda^(2K/L) * log(L+1)) for uniform conditional full-label AWGN service, optimizing over every group partition and local amplitude allocation. A broader classical converse requires Omega(K/log K) complex payload observations at polynomial signal energy. For near-minimum-energy local geometric hierarchies with average group size c log_lambda L, c > 1/2, a single arbitrarily correlated phase-description encoder still requires Theta(K log K) bits, even with unequal groups and energies. General-signature bit optimality and sharp constants remain unresolved. Additional results include an exact convex optimization of a retained genie reliability relaxation, a low-error balancing theorem, a rationally certified high-error counterexample to extending that theorem without its assumptions, and a noisy-pilot construction that counts received acquisition energy, pilot observations, confidence failures, and coherence/amortization requirements. One-vector training costs K additional pilot observations; ideal payload-dimension savings are not presented as total deployment savings. Research_5_Unified_Theory_Package.zip is the complete Research #5 release: modular LaTeX, figures, standard-library certificate checkers, reproducible results, planner, pinned optional dependencies, literature and technical audits, provenance, preserved manuscript versions, and explicitly identified source context from Researches #1–#4. Separate files provide convenient access to the current paper and central supplementary materials. SHA256SUMS.txt covers 257 archive entries. The independent-in-code v1.7 checker passes normally and under Python optimization with identical outputs; original and finite-QAM certificates were replayed. These are internal/adversarial checks, not external peer review or independent external replication. No new coded-link frames or physical trials were performed. The work does not establish net supply-energy savings, improved coverage, commercial superiority, historical priority, or a complete architecture-independent joint frontier. AI assistance was used in derivation, proof exploration, writing, and code preparation. No additional reuse license is asserted for the author's work; any third-party material retains its existing terms.